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<b>let </b><font color="Maroon" title="c1">k</font> be  <a href="ordinal1.html#V7" title="ORDINAL1:attr.7">natural</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">S</font> being  <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Olive" title="b1">X</font><br/> for <font color="Olive" title="b3">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Olive" title="b1">X</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <br/> for <font color="Olive" title="b4">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b2">S</font>  st <font color="Olive" title="b4">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b3">f</font> &amp; <font color="Olive" title="b3">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b4">E</font> holds <br/><span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Olive" title="b3">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b4">E</font></span><br/><b>let </b><font color="Maroon" title="c2">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X</font><br/> for <font color="Olive" title="b2">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <br/> for <font color="Olive" title="b3">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b3">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b2">f</font> &amp; <font color="Olive" title="b2">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b3">E</font> holds <br/><span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Olive" title="b2">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b3">E</font></span><br/><b>let </b><font color="Maroon" title="c3">S</font> be   <a href="prob_1.html#NM3" title="PROB_1:NM.3">SigmaField</a> of <font color="Maroon" title="c2">X</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> <br/> for <font color="Olive" title="b2">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font>  st <font color="Olive" title="b2">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Olive" title="b1">f</font> &amp; <font color="Olive" title="b1">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b2">E</font> holds <br/><span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Olive" title="b1">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b2">E</font></span><br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">X</font>, <a href="numbers.html#K7" title="NUMBERS:func.7">ExtREAL</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">E</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font>  st <font color="Olive" title="b1">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</font> holds <br/><span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Olive" title="b1">E</font></span><br/><b>let </b><font color="Maroon" title="c5">E</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c3">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font> implies <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font> )</span><br/>









<b>reconsider </b><font color="Maroon" title="c6">k1</font> = <font color="Maroon" title="c1">k</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>assume </b><a NAME="E2:30"/><i><font color="Green" title="E18">A1</font></i>: 
( <font color="Maroon" title="c5">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c4">f</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font></span><br/>

<a NAME="E3:30"/><i><font color="Green" title="E19">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span>
 
<b>by </b><i><a class="ref" href="mesfunc7.html#D5" target="_self" title="MESFUNC7:def.5">Def5</a></i>;<br/>
<a NAME="E4:30"/><b>then </b><i><font color="Green" title="E20">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c4">f</font>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1"/>
( <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_1"><b>suppose </b></a><a NAME="E1:30_1_1"/><i><font color="Green" title="E21">A4</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font></span><br/><div class="add"><a NAME="E2:30_1_1"/><i><font color="Green" title="E22">A5</font></i>: 
for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> holds <br/><font color="Maroon" title="c5">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">less_dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Olive" title="b1">r</font></span>)</span></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c3">S</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> implies <font color="Maroon" title="c5">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">less_dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c3">S</font> )</span><br/>

<b>assume </b><a NAME="E1:30_1_1_1"/><i><font color="Green" title="E23">A6</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">less_dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c3">S</font></span><br/>

<a NAME="E2:30_1_1_1"/>
<font color="Maroon" title="c5">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="mesfunc1.html#K19" title="MESFUNC1:func.19">less_dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K19" title="MESFUNC1:func.19">less_dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:30_1_1_1_1"/><i><font color="Green" title="E24">A7</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K19" title="MESFUNC1:func.19">less_dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span></span><br/>

<a NAME="E2:30_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E2:30"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="mesfunc7.html#E4:30"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="mesfunc7.html#D5" target="_self" title="MESFUNC7:def.5">Def5</a></i>;<br/>
<a NAME="E3:30_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E1:30_1_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="mesfunc7.html#E1:30_1_1_1"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="mesfunc7.html#T6" target="_self" title="MESFUNC7:th.6">Th6</a>, <a class="ref" href="finseq_2.html#T72" title="FINSEQ_2:th.72">FINSEQ_2:72</a></i>;<br/>
<b>hence </b><a NAME="E4:30_1_1_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K19" title="MESFUNC1:func.19">less_dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E2:30"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="mesfunc7.html#E4:30"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="mesfunc7.html#E1:30_1_1_1_1"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="mesfunc1.html#D12" title="MESFUNC1:def.12">MESFUNC1:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:30_1_1_1"/><b>then </b>
<font color="Maroon" title="c5">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">less_dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<b>hence </b><a NAME="E4:30_1_1_1"/>
<font color="Maroon" title="c5">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">less_dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c3">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E3:30_1_1"/><i><font color="Green" title="E23">A8</font></i>: 
<font color="Maroon" title="c5">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E2:30"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="mesfunc7.html#E3:30"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/><a NAME="E4:30_1_1"/>
for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon" title="c5">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">less_dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Olive" title="b1">r</font></span>)</span></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c3">S</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">less_dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c3">S</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_1_2_1"/><i><font color="Green" title="E24">A9</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">less_dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c3">S</font></span><br/><a NAME="E2:30_1_1_2_1"/>
<font color="Maroon" title="c5">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:30_1_1_2_1_1"/><i><font color="Green" title="E25">A10</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span></span><br/>

<a NAME="E2:30_1_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E2:30"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="mesfunc7.html#E4:30"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="mesfunc7.html#D5" target="_self" title="MESFUNC7:def.5">Def5</a></i>;<br/>
<a NAME="E3:30_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E1:30_1_1"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="mesfunc7.html#E1:30_1_1_2_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="ref" href="mesfunc7.html#T6" target="_self" title="MESFUNC7:th.6">Th6</a>, <a class="ref" href="finseq_2.html#T72" title="FINSEQ_2:th.72">FINSEQ_2:72</a></i>;<br/>
<b>hence </b><a NAME="E4:30_1_1_2_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E2:30"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="mesfunc7.html#E4:30"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="mesfunc7.html#E1:30_1_1_2_1_1"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="mesfunc1.html#D15" title="MESFUNC1:def.15">MESFUNC1:def 15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E3:30_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c5">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/><a NAME="E4:30_1_1_2_1"/><b>then </b>
<font color="Maroon" title="c5">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">less_dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font> <a href="measure1.html#K3" title="MEASURE1:func.3">\</a> <font color="Maroon" title="c5">E</font>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E3:30_1_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="mesfunc1.html#T21" title="MESFUNC1:th.21">MESFUNC1:21</a></i>;<br/><b>hence </b><a NAME="E5:30_1_1_2_1"/>
<font color="Maroon" title="c5">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">less_dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c3">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:30_1_1_2"/>
<font color="Maroon" title="c5">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K19" title="MESFUNC1:func.19">less_dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c3">S</font>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E2:30_1_1"><i><font color="Green" title="E22">A5</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:30_1_1"/>
<span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font>
 <b>by </b><i><a class="ref" href="mesfunc1.html#D17" title="MESFUNC1:def.17">MESFUNC1:def 17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2"><b>suppose </b></a><a NAME="E1:30_1_2"/><i><font color="Green" title="E21">A11</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font></span><br/><div class="add"><a NAME="E2:30_1_2"/><b>then </b><i><font color="Green" title="E22">A12</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E3:30_1_2"/><i><font color="Green" title="E23">A13</font></i>: 
<span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="txt" href="mesfunc7.html#E1:30_1_2"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="xcmplx_1.html#T88" title="XCMPLX_1:th.88">XCMPLX_1:88</a></i>;<br/><a NAME="E4:30_1_2"/><i><font color="Green" title="E24">A14</font></i>: 
for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">r</font> holds <br/> <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> implies  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:30_1_2_1"/><i><font color="Green" title="E25">A15</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span></span><br/>

<a NAME="E2:30_1_2_1"/><i><font color="Green" title="E26">A16</font></i>: 
 <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:30_1_2_1_1"/><i><font color="Green" title="E27">A17</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span></span><br/>

<a NAME="E2:30_1_2_1_1"/><b>then </b><i><font color="Green" title="E28">A18</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D15" title="MESFUNC1:def.15">MESFUNC1:def 15</a></i>;<br/>
<a NAME="E3:30_1_2_1_1"/><b>then </b>
<span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/>
<a NAME="E4:30_1_2_1_1"/><b>then </b><i><font color="Green" title="E29">A19</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="ref" href="extreal2.html#T51" title="EXTREAL2:th.51">EXTREAL2:51</a></i>;<br/>
<a NAME="E5:30_1_2_1_1"/><i><font color="Green" title="E30">A20</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E3:30"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="mesfunc7.html#E2:30_1_2_1_1"><i><font color="Green" title="E28">A18</font></i></a>, <a class="ref" href="mesfunc7.html#D5" target="_self" title="MESFUNC7:def.5">Def5</a></i>;<br/>
<div><i><font color="Green" title="E31">A21</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_2_1_1_1"/>
<span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span><br/><a NAME="E2:30_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 <b>by </b><i><a class="txt" href="mesfunc7.html#E1:30_1_2"><i><font color="Green" title="E21">A11</font></i></a>, <a class="ref" href="mesfunc7.html#T13" target="_self" title="MESFUNC7:th.13">Th13</a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><b>hence </b><a NAME="E3:30_1_2_1_1_1"/>
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E5:30_1_2_1_1"><i><font color="Green" title="E30">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T3" title="XXREAL_0:th.3">XXREAL_0:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:30_1_2_1_1_2"/>
<span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span><br/><b>then reconsider </b><font color="Maroon" title="c9">fx</font> = <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  <b>by </b><i><a class="txt" href="mesfunc7.html#E4:30_1_2_1_1"><i><font color="Green" title="E29">A19</font></i></a>, <a class="ref" href="supinf_2.html#T2" title="SUPINF_2:th.2">SUPINF_2:2</a></i>;<br/><a NAME="E3:30_1_2_1_1_2"/><i><font color="Green" title="E32">A22</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">fx</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E5:30_1_2_1_1"><i><font color="Green" title="E30">A20</font></i></a>, <a class="ref" href="mesfunc7.html#T11" target="_self" title="MESFUNC7:th.11">Th11</a></i>;<br/><a NAME="E4:30_1_2_1_1_2"/><i><font color="Green" title="E33">A23</font></i>: 
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E1:30_1_2_1_1"><i><font color="Green" title="E27">A17</font></i></a>, <a class="ref" href="mesfunc1.html#D15" title="MESFUNC1:def.15">MESFUNC1:def 15</a></i>;<br/><a NAME="E5:30_1_2_1_1_2"/><i><font color="Green" title="E34">A24</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E1:30_1_2_1"><i><font color="Green" title="E25">A15</font></i></a>, <a class="ref" href="power.html#T39" title="POWER:th.39">POWER:39</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c10">R</font> = <font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E7:30_1_2_1_1_2"/>
<font color="Maroon" title="c10">R</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c6">k1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E1:30_1_2_1"><i><font color="Green" title="E25">A15</font></i></a>, <a class="ref" href="power.html#T38" title="POWER:th.38">POWER:38</a></i>;<br/><a NAME="E8:30_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E35">A25</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">fx</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E2:30_1_2"><i><font color="Green" title="E22">A12</font></i></a>, <a class="txt" href="mesfunc7.html#E3:30_1_2"><i><font color="Green" title="E23">A13</font></i></a>, <a class="txt" href="mesfunc7.html#E4:30_1_2_1_1_2"><i><font color="Green" title="E33">A23</font></i></a>, <a class="txt" href="mesfunc7.html#E5:30_1_2_1_1_2"><i><font color="Green" title="E34">A24</font></i></a>, <a class="ref" href="holder_1.html#T3" title="HOLDER_1:th.3">HOLDER_1:3</a></i>;<br/><b>thus </b><a NAME="E9:30_1_2_1_1_2"/>
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E3:30_1_2_1_1_2"><i><font color="Green" title="E32">A22</font></i></a>, <a class="txt" href="mesfunc7.html#E8:30_1_2_1_1_2"><i><font color="Green" title="E35">A25</font></i></a>, <a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:30_1_2_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E3:30"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="mesfunc7.html#E2:30_1_2_1_1"><i><font color="Green" title="E28">A18</font></i></a>, <a class="txt" href="mesfunc7.html#E6:30_1_2_1_1"><i><font color="Green" title="E31">A21</font></i></a>, <a class="ref" href="mesfunc1.html#D15" title="MESFUNC1:def.15">MESFUNC1:def 15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E3:30_1_2_1"/>
 <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:30_1_2_1_2"/><i><font color="Green" title="E27">A26</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span></span><br/>

<a NAME="E2:30_1_2_1_2"/><b>then </b><i><font color="Green" title="E28">A27</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="mesfunc1.html#D15" title="MESFUNC1:def.15">MESFUNC1:def 15</a></i>;<br/>
<a NAME="E3:30_1_2_1_2"/><b>then </b><i><font color="Green" title="E29">A28</font></i>: 
<span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E3:30"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/>
<a NAME="E4:30_1_2_1_2"/><b>then </b><i><font color="Green" title="E30">A29</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="ref" href="extreal2.html#T51" title="EXTREAL2:th.51">EXTREAL2:51</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_2_1_2_1"/>
( <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  or <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1_2_1_1"><b>suppose </b></a><a NAME="E1:30_1_2_1_2_1_1"/>
<span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:30_1_2_1_2_1_1"/><b>then </b>
 <a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p2">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T3" title="XXREAL_0:th.3">XXREAL_0:3</a></i>;<br/><b>hence </b><a NAME="E3:30_1_2_1_2_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E3:30"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="mesfunc7.html#E2:30_1_2_1_2"><i><font color="Green" title="E28">A27</font></i></a>, <a class="ref" href="mesfunc1.html#D15" title="MESFUNC1:def.15">MESFUNC1:def 15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_1_2_1_2"><b>suppose </b></a><a NAME="E1:30_1_2_1_2_1_2"/>
<span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="supinf_1.html#K1" title="SUPINF_1:func.1">+infty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span></span><br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c9">fx</font> = <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a>  <b>by </b><i><a class="txt" href="mesfunc7.html#E4:30_1_2_1_2"><i><font color="Green" title="E30">A29</font></i></a>, <a class="ref" href="supinf_2.html#T2" title="SUPINF_2:th.2">SUPINF_2:2</a></i>;<br/><a NAME="E3:30_1_2_1_2_1_2"/><i><font color="Green" title="E31">A30</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E2:30_1_2"><i><font color="Green" title="E22">A12</font></i></a>, <a class="ref" href="real_2.html#T127" title="REAL_2:th.127">REAL_2:127</a></i>;<br/><a NAME="E4:30_1_2_1_2_1_2"/><i><font color="Green" title="E32">A31</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E1:30_1_2_1_2"><i><font color="Green" title="E27">A26</font></i></a>, <a class="ref" href="mesfunc1.html#D15" title="MESFUNC1:def.15">MESFUNC1:def 15</a></i>;<br/><a NAME="E5:30_1_2_1_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E2:30_1_2_1_2"><i><font color="Green" title="E28">A27</font></i></a>, <a class="ref" href="mesfunc7.html#D5" target="_self" title="MESFUNC7:def.5">Def5</a></i>;<br/><a NAME="E6:30_1_2_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">fx</font> <a href="newton.html#K3" title="NEWTON:func.3">|^</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="ref" href="mesfunc7.html#T11" target="_self" title="MESFUNC7:th.11">Th11</a></i>;<br/><a NAME="E7:30_1_2_1_2_1_2"/><b>then </b><i><font color="Green" title="E33">A32</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">fx</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <font color="Maroon" title="c6">k1</font>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E4:30_1_2_1_2_1_2"><i><font color="Green" title="E32">A31</font></i></a></i>;<br/><a NAME="E8:30_1_2_1_2_1_2"/>
<font color="Maroon" title="c7">r</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/><a NAME="E9:30_1_2_1_2_1_2"/><b>then </b><i><font color="Green" title="E34">A33</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">fx</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E1:30_1_2_1"><i><font color="Green" title="E25">A15</font></i></a>, <a class="txt" href="mesfunc7.html#E3:30_1_2_1_2_1_2"><i><font color="Green" title="E31">A30</font></i></a>, <a class="txt" href="mesfunc7.html#E7:30_1_2_1_2_1_2"><i><font color="Green" title="E33">A32</font></i></a>, <a class="ref" href="holder_1.html#T3" title="HOLDER_1:th.3">HOLDER_1:3</a></i>;<br/><a NAME="E10:30_1_2_1_2_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">k</font>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E1:30_1_2"><i><font color="Green" title="E21">A11</font></i></a></i>;<br/><a NAME="E11:30_1_2_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">fx</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">fx</font> <a href="power.html#K4" title="POWER:func.4">to_power</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K3" title="NAT_1:func.3">*</a> 1</span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E3:30_1_2_1_2"><i><font color="Green" title="E29">A28</font></i></a>, <a class="txt" href="mesfunc7.html#E3:30_1_2_1_2_1_2"><i><font color="Green" title="E31">A30</font></i></a>, <a class="ref" href="extreal2.html#T51" title="EXTREAL2:th.51">EXTREAL2:51</a>, <a class="ref" href="holder_1.html#T2" title="HOLDER_1:th.2">HOLDER_1:2</a></i>;<br/><a NAME="E12:30_1_2_1_2_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">fx</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">fx</font>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E3:30_1_2"><i><font color="Green" title="E23">A13</font></i></a>, <a class="ref" href="power.html#T30" title="POWER:th.30">POWER:30</a></i>;<br/><b>hence </b><a NAME="E13:30_1_2_1_2_1_2"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E3:30"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="mesfunc7.html#E2:30_1_2_1_2"><i><font color="Green" title="E28">A27</font></i></a>, <a class="txt" href="mesfunc7.html#E9:30_1_2_1_2_1_2"><i><font color="Green" title="E34">A33</font></i></a>, <a class="ref" href="mesfunc1.html#D15" title="MESFUNC1:def.15">MESFUNC1:def 15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E4:30_1_2_1"/>
 <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E2:30_1_2_1"><i><font color="Green" title="E26">A16</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E5:30_1_2"/><i><font color="Green" title="E25">A34</font></i>: 
<span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E2:30"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="mesfunc2.html#T29" title="MESFUNC2:th.29">MESFUNC2:29</a></i>;<br/><a NAME="E6:30_1_2"/>
for <font color="Olive" title="b1">r</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>  holds  <font color="Maroon" title="c5">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Olive" title="b1">r</font></span>)</span></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c3">S</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c7">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c3">S</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:30_1_2_2_1"/>
( <font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:30_1_2_2_1_1"/><i><font color="Green" title="E26">A35</font></i>: 
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c3">S</font></span><br/><div class="add"><a NAME="E2:30_1_2_2_1_1"/>
<font color="Maroon" title="c5">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1_2_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font> or <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:30_1_2_2_1_1_1"/><i><font color="Green" title="E27">A36</font></i>: 
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">E</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span></span><br/>

<a NAME="E2:30_1_2_2_1_1_1"/><b>then </b><i><font color="Green" title="E28">A37</font></i>: 
<span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span> <a href="mesfunc7.html#K6" title="MESFUNC7:func.6">|^</a> <font color="Maroon" title="c1">k</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E2:30"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="mesfunc7.html#E4:30"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="mesfunc7.html#D5" target="_self" title="MESFUNC7:def.5">Def5</a></i>;<br/>
<a NAME="E3:30_1_2_2_1_1_1"/>
<span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="extreal1.html#K4" title="EXTREAL1:func.4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font></span>)</span></span><a href="extreal1.html#K4" title="EXTREAL1:func.4">.|</a></span>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E2:30"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="mesfunc7.html#E3:30"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="mesfunc7.html#E4:30"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="mesfunc7.html#E1:30_1_2_2_1_1_1"><i><font color="Green" title="E27">A36</font></i></a>, <a class="ref" href="mesfunc1.html#D10" title="MESFUNC1:def.10">MESFUNC1:def 10</a></i>;<br/>
<a NAME="E4:30_1_2_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span> <a href="supinf_2.html#K12" title="SUPINF_2:func.12">.</a> <font color="Maroon" title="c8">x</font>
 
<b>by </b><i><a class="txt" href="mesfunc7.html#E1:30_1_2_2_1_1"><i><font color="Green" title="E26">A35</font></i></a>, <a class="txt" href="mesfunc7.html#E2:30_1_2_2_1_1_1"><i><font color="Green" title="E28">A37</font></i></a>, <a class="ref" href="mesfunc7.html#T12" target="_self" title="MESFUNC7:th.12">Th12</a>, <a class="ref" href="extreal2.html#T51" title="EXTREAL2:th.51">EXTREAL2:51</a></i>;<br/>
<b>hence </b><a NAME="E5:30_1_2_2_1_1_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p1">(<span class="default"><span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E2:30"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="mesfunc7.html#E4:30"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="mesfunc7.html#E1:30_1_2_2_1_1_1"><i><font color="Green" title="E27">A36</font></i></a>, <a class="ref" href="mesfunc1.html#D15" title="MESFUNC1:def.15">MESFUNC1:def 15</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><a NAME="E3:30_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c5">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/><b>hence </b><a NAME="E4:30_1_2_2_1_1"/>
<font color="Maroon" title="c5">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c3">S</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="30_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:30_1_2_2_1_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">r</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c3">S</font></span><br/><div class="add"><a NAME="E2:30_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c5">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p2"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">r</font> <a href="power.html#K3" title="POWER:func.3">to_power</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c1">k</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E4:30_1_2"><i><font color="Green" title="E24">A14</font></i></a></i>;<br/><b>hence </b><a NAME="E3:30_1_2_2_1_2"/>
<font color="Maroon" title="c5">E</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="mesfunc1.html#K22" title="MESFUNC1:func.22">great_eq_dom</a> <span class="p2">(<span class="default"><span class="p3"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="measure6.html#K1" title="MEASURE6:func.1">R_EAL</a> <font color="Maroon" title="c7">r</font></span>)</span></span>)</span> <a href="mesfunc1.html#NR1" title="MESFUNC1:NR.1">is_measurable_on</a> <font color="Maroon" title="c3">S</font>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E2:30"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="mesfunc7.html#E3:30"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="mesfunc7.html#E4:30"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="mesfunc7.html#E5:30_1_2"><i><font color="Green" title="E25">A34</font></i></a>, <a class="ref" href="mesfunc1.html#T31" title="MESFUNC1:th.31">MESFUNC1:31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E7:30_1_2"/>
<span class="p1"><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">|.</a><span class="default"><font color="Maroon" title="c4">f</font></span><a href="mesfunc7.html#K1" title="MESFUNC7:func.1">.|</a></span> <a href="mesfunc7.html#K7" title="MESFUNC7:func.7">|^</a> <font color="Maroon" title="c1">k</font> <a href="mesfunc1.html#R1" title="MESFUNC1:pred.1">is_measurable_on</a> <font color="Maroon" title="c5">E</font>
 <b>by </b><i><a class="txt" href="mesfunc7.html#E2:30"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="mesfunc7.html#E4:30"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="mesfunc1.html#T31" title="MESFUNC1:th.31">MESFUNC1:31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
